Among the most widely-used instruments for measuring surface topographies are interferometers, which use the wave nature of light to map variations in surface height with a high degree of accuracy. It is generally accepted in the art that the most accurate interferometers are based on the principle of phase shifting. Modern phase-shifting interferometers are typically comprised of an optical system, an electronic imaging system, a computer-based or otherwise automated controller, and means for introducing a reference phase shift.
Phase shifting interferometry (PSI), for example, is described in detail in Chapter 14 of a book entitled Optical Shop Testing, edited by Daniel Malacara (Wiley, New York, 1992). Briefly described, PSI typically involves electronic storage of intensity patterns measured for a sequence of three or more reference phase shifts. These stored intensity patterns are then analyzed, as by a computer-based digital signal processor, to recover the original wavefront phase through analysis of the variations of intensity as a function of phase shift. When such PSI-based instruments are properly adjusted, they are capable of measuring surface topography with a resolution on the order of one-thousandth the wavelength of light.
An interferometric wavefront sensor employing phase-shifting interferometry typically consists of a light source that is split into two wavefronts, namely, reference and test wavefronts, that are recombined after traveling through different paths. The relative phase difference between the two wavefronts is manifested as a two-dimensional intensity pattern known as an interferogram. Phase-shift interferometers typically have an element in the path of the reference or the test wavefront which introduces three or more known phase steps or phase shifts. By detecting the intensity pattern with a detector, at each of the phase shifts, the phase distribution of the object wavefront can be quantitatively calculated independent of the relative energy in the reference wavefront or object wavefront.
Phase shifting of a light beam can either be accomplished by sequentially introducing a phase step (temporal phase shifting) or by splitting the beam into parallel channels for simultaneous phase steps (spatial phase shifting). Spatial phase shifting achieves data acquisition in a time period that is several orders of magnitude less than temporal phase shifting, thus offering immunity to vibration.
U.S. Pat. No. 7,230,717, issued Jun. 12, 2007, describes how a pixilated phase-mask can be used as an interferometer to measure optical path-length differences at high-sped, with a single detector array. Portions of this Patent are described below with reference to FIGS. 1 and 2.
With reference to FIG. 1, a complete measurement system 10 is shown. The system consists of a polarization interferometer 12 that generates a reference wavefront R and a test wavefront T having orthogonal polarization states (which can be linear as well as circular) with respect to each other; a pixilated phase mask (PPM) 14 that introduces an effective phase-delay between the reference and test wavefronts at each pixel and, subsequently, interferes the transmitted light; and a detector array 16 that converts the optical intensity sensed at each pixel to an electrical charge. The measurement system 10 also includes an amplifier 18 that converts the electrical charge to a voltage, a digitization circuit 20 that converts the voltage to a digital bit stream, a computer 22 that processes the digital bit stream to calculate optical phase difference, and a display 24 that conveys the result in visible form and permits user interaction.
The PPM 14 has an effective pixel pitch or spacing that is identical to, or an integer multiple of, the pixel pitch of the detector array 16. Additionally, the PPM 14 is rotationally and axially aligned with respect to the detector array 16, so that the effective pixels of the pixilated phase-mask and the pixels of the detector array are substantially coincident across the entire surface of each.
The PPM 14 and detector array 16 may be located in the same image plane, or positioned in conjugate image planes. The PPM can be directly deposited over, or affixed onto detector array 16, or can be mechanically registered and separated by a small gap.
A complete measurement system 50 is illustrated in FIG. 2, wherein the pixelated phase-mask (PPM) 14 is used in conjunction with a Twyman-Green interferometer 52. A linearly polarized beam L from a light source 54 is combined with a half-wave plate 56 to produce a linearly polarized beam of desired polarization angle directed to a polarizing beam splitter 58, which in turn generates a reference beam directed toward a reference surface 60 and a test beam directed toward a test surface 62. Both beams are linearly polarized along orthogonal axes. Quarter-wave plates 64 and 66 are used to rotate the test and reference beams, T and R, after reflection while retaining their mutually orthogonal linear polarization states, so that they may be transmitted through and reflected from beam splitter 58, respectively, toward relay optics 26, 28 and 30. A coupling lens 65 is used in combination with test surface 62 to return a substantially collimated test beam T.
As one skilled in the art would readily understand, the pixelated phase-mask (PPM) can similarly be combined in a plurality of other systems designed to carry out particular types of real-time measurement, such as with a Fizeau interferometer, a microscope profilometer, a wave-front sensor, and a strain sensor.
The effective phase-shift of each pixel of PPM 14 can have any spatial distribution; however, it is desirable to have a regularly repeating pattern. One example for the PPM is an arrangement in which neighboring pixels are in quadrature or out-of-phase with respect to each other; that is, there is a ninety-degree, or one hundred eighty degree relative phase shift between neighboring pixels. Many algorithms exist in the art for calculating phase from sampled data in quadrature.
FIG. 3 illustrates one possible way of arranging the PPM and the detector pixels and for processing the measured data. The measured array 300 represents the signal measured at each sensor pixel. The capital letters A, B, C and D represent different transfer functions resulting by the filtering from the pixelated phase-mask (the transfer functions are shown labeled 301, 302, . . . and 315.)
The signal measured at each sensor pixel is given by its transfer function, the phase-difference between the reference and test beams, and the amplitude of each beam. For example, one possible configuration is the following:
                              A          ⁡                      (                          x              ,              y                        )                          =                              1            2                    ⁢                      (                                          I                r                            +                              I                s                            +                              2                ⁢                                                                            I                      r                                        ⁢                                          I                      s                                                                      ⁢                                  cos                  ⁡                                      (                                          Δϕ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                              )                                                                        )                                              (        1        )                                          B          ⁡                      (                          x              ,              y                        )                          =                              1            2                    ⁢                      (                                          I                r                            +                              I                s                            +                              2                ⁢                                                                            I                      r                                        ⁢                                          I                      s                                                                      ⁢                                  cos                  ⁡                                      (                                                                  Δϕ                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                                              π                        2                                                              )                                                                        )                                              (        2        )                                          C          ⁡                      (                          x              ,              y                        )                          =                              1            2                    ⁢                      (                                          I                r                            +                              I                s                            +                              2                ⁢                                                                            I                      r                                        ⁢                                          I                      s                                                                      ⁢                                  cos                  ⁡                                      (                                                                  Δϕ                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                      π                                        )                                                                        )                                              (        3        )                                          D          ⁡                      (                          x              ,              y                        )                          =                              1            2                    ⁢                      (                                          I                r                            +                              I                s                            +                              2                ⁢                                                                            I                      r                                        ⁢                                          I                      s                                                                      ⁢                                  cos                  ⁡                                      (                                                                  Δϕ                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                                                                        3                          ⁢                          π                                                2                                                              )                                                                        )                                              (        4        )            where Ir(x, y) and Is(x, y) are the intensities of the reference and test wavefronts, R and T, at each x, y coordinate in the image, respectively; and Δφ(x, y) is the optical path difference between the reference and test wavefronts.
Multiple interferograms can thus be synthesized by combining pixels with like transfer functions. Referring to FIG. 3, the pixels with transfer functions equal to A can be combined into an interferogram 350. For example, measured intensity pixel 301 is mapped to interferogram pixel 351; intensity pixel 303 to interferogram pixel 352; intensity pixel 312 to interferogram pixel 353; intensity pixel 314 to interferogram pixel 354; and so on. The resulting interferogram 350 is a continuous fringe map that opticians are accustomed to viewing for alignment, which can be displayed on a screen in real-time. The B, C, and D pixels can be similarly combined to produce corresponding interferograms.
The resulting interferograms have a total number of pixels equal to (n×m)/N, where n and m are the numbers of pixels in the detector array in the x and y directions, respectively, and N is the number of different discrete phase-shift elements in the pixelated phase mask 10. In the example of FIG. 3, N is equal to four. The resulting four interferograms can be processed by a variety of algorithms that are well-known in the art for calculating phase difference.
FIG. 4 illustrates four phase-shifted interferograms, generally designated as 406, which are produced by parsing the pixel-wise phase shift pattern 405. It will be appreciated that A, B, C and D shown as pixels may each be based on a single pixel, or a set of multiple pixels.
The phase shift pattern is produced by a measurement system, designated as 400. The measurement system includes a single mode laser 401, a camera 402, interferometer 403 and test mirror 404. The pixels of camera 402 produces the pixel-wise phase shift pattern 405. More detail of measurement system 400 may be obtained by referring to the description of system 50, shown in FIG. 2.
The resulting four interferograms can be processed by a variety of algorithms that are known in the art for calculating phase difference and modulation index. For example, a possible implementation for measuring phase difference is a simple four-bucket algorithm, as follows:φ(x,y)=arctangent[(A−C)/(D−B)]  (5)                where φ(x, y) is the phase, and                    A, B, C and D are taken from adjacent neighboring pixels in FIG. 4.                        
Prior art phase shifting interferometers, which use the four-bucket algorithm, suffer from a measurement defect known as fringe print through. This is a common problem in a four-bucket algorithm, as will as other algorithms. The measurement defect may typically be reduced by taking many measurements of the test hardware, with intentional vibration added to the hardware under test. Adding vibration, however, is not always adequate if the induced vibration is not large enough, random enough, or is not well prescribed. Thus, unacceptable errors may typically result.
The present invention, as will be described, provides a method and an algorithm for reducing the fringe print through and achieving improved results in all simultaneous phase shifting measurements. Improved results may be seen in simultaneous phase shifting (or in polarization phase shifting) wavefront sensors, piston sensors, or interferometers for providing various applications, including system wavefront testing, component surface testing, and speckle testing.